Given a radius, can you calculate a surface area? Given an area and a "std." atmospheric pressure can you calculate a total force? Given that force and an "average" value for acceleration of gravity at the earth's surface, can you calculate anything else of interest?

Easy (this is probably what you want to do)- determine the difference in the acceleration due to gravity at the high and low ends of the atmosphere. This will (probably) allow you to make a very nice simplifying assumption so you can get a good approximation quickly and easily using the surface air pressure, the acceleration of gravity, and the surface area of the earth.

Medium - Integrate by shells assuming that the earth is spherical, and the temperature of the atmosphere is constant. Remember that the density is proportional to the pressure.

Hard - Integrate but account for the fact that the earth is a spinning elipsoid and for temperature with respect to lattititude and altititude.